Vector Fields in 2 and 3 Dimensions

Since a vector field assigns vectors to points, they can be written as 3 dimensional vector-valued functions:

F ( x,y,z ) =

If  for all ( x,y,z ) and M,N do not depend on z , then the vector field is 2-dimensional, and we write either

F ( x,y ) =        or       F ( x,y ) =

For example, the vector field F ( x,y ) =   assigns the vector  to the point ( x,y ).  To illustrate, the vector F =  is assigned to the point (4,3).

Let's look at this vector field with Maple .  To begin with, we must define coordinates with the SetCoordinates  command.  A vector field is then defined with the VectorField command in terms of the coordinate variables defined in SetCoordinates.  

>	SetCoordinates(cartesian[x,y]); F:=VectorField(<-y,x>);

>

The fieldplot command is then used to graph the vector-field.

>	fieldplot(F,x=-1..1,y=-1..1,color=red,arrows=slim);

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To plot 2-dimensional vector fields with Maple , we use the fieldplot3d  command. First, however, we must change our coordinates to 3-dimensional cartesian.

>	SetCoordinates(cartesian[x,y,z]): fieldplot3d([-y,x,z],x=-5..5,y=-5..5,z=-5..5,arrows=THICK,color=red,grid=[5,5,5]);

>

A vector field  is often studied with the concept of a flow , where a set of curves is a flow  of a 2-dimensional vector field F ( x,y ) =  if for every point ( x, y) in the plane, there is a curve r ( t ) in the flow that satisfies

 = F    and r (0) = ( x,y )

For example, the set of concentric circles centered at the origin is the flow of the 2-dimensional vector field F ( x,y ) = , as is shown in the animated figure below:  

(commands used to create the animation above)